US 7720661 B2
A method for a geometry of a lateral comb drive for an in-plane, electrostatic force feedback, closed-loop, micromachined accelerometer or closed-loop Coriolis rate gyroscope device, or closed-loop capacitive pressure or force measuring device. When vibration is applied to the device, the error in the time-average output, which is vibration rectification error, due to this input vibration is minimized or eliminated. The geometry resulting from practice of the present invention is space-efficient because drive force is maximized while vibration rectification is minimized or eliminated.
1. A method for making a microelectromechanical system (MEMS) electrostatic comb-drive device, the method comprising:
constructing a finite element model of an initial comb tooth geometric unit;
selecting an initial comb tooth overlap dimension;
performing a finite element analysis calculation of capacitance for both the selected initial overlap dimension and for each of one or more different overlap dimensions adjacent to the initial overlap dimension;
constructing a model of a relationship of capacitance versus overlap dimension;
applying a second derivative test to solve for an inflection point in the relationship of capacitance versus overlap dimension;
selecting an overlap dimension corresponding to the inflection point; and
making a MEMS electrostatic comb-drive by orienting a movable proof mass having comb teeth to a stationary comb tooth frame, so that the comb teeth and the stationary comb tooth frame overlap by the selected overlap dimension.
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initially adjusting a position of the proof mass relative to the frame to a position of non-zero differential capacitance;
vibrating the adjusted proof mass relative to the frame along an input axis;
additionally adjusting the position of the vibrating proof mass relative to the frame to a position where fluid forces operating on the proof mass are balanced relative to the comb tooth geometry, thereby calibrating the MEMS device.
10. The method of
11. The method of
vibrating the device along an input axis; and
balancing fluid forces operating on the vibrating proof mass with the actual comb tooth geometry.
12. The method of
13. The method of
wherein adjusting a position of the proof mass relative to the frame to a different relative position where fluid forces operating on the proof mass are balanced with the actual comb tooth geometry further comprises, after initially adjusting the relative position of the proof mass to the zero rectification position and during vibrating the device, additionally adjusting the position of the proof mass relative to the frame.
The present invention relates to micromachined sensor devices and methods, and in particular to electrostatic comb-drive, closed-loop, in-plane, micromachined, capacitive accelerometers, Coriolis rate gyroscopes, and pressure and force measuring devices.
Microelectromechanical system (MEMS) capacitive electrostatic comb-drive, closed-loop, in-plane, micromachined, capacitive pick-off accelerometer devices, closed-loop Coriolis rate gyroscope devices, and closed-loop capacitive pressure and force measuring devices are generally well-known. In particular, silicon-based, micromachined accelerometers are displacing accelerometers of more mature architectures in current applications, and are creating new markets where the advantages of small size and low cost are enabling qualities. One critical area of performance that poses a major challenge for MEMS capacitive accelerometers is vibration rectification. Vibration rectification is the change in the time-average accelerometer output due to input vibration. Vibration rectification manifests as an apparent change in the DC acceleration when none is being experienced.
Current MEMS capacitive accelerometers, Coriolis rate gyroscope devices, and closed-loop capacitive force measuring devices have very poor vibration rectification performance. For example, an input vibration of 10 Grms along the input axis of a known electrostatic comb-drive MEMS capacitive pick-off accelerometer is able to change the average output by as much as 0.1 g's. This large vibration rectification makes these accelerometers unsuitable for current tactical and navigation-grade applications.
In a closed-loop capacitive pick-off accelerometer, rectification error is driven by several sources. For example, rebalance force is not linear relative to the voltage applied to the electrostatic comb-drive. The rebalancing force is proportional to the square of the applied voltage difference between sets of interacting moveable and fixed comb teeth. There are several well-known ways to accomplish linearization of the applied voltage. For example, a square root function can be placed in the feedback loop. The various methods of linearizing this relationship, however, are not relevant to the present invention.
Scale factor may have asymmetry in a closed-loop capacitive pick-off accelerometer. That is, the scale factor in the positive input direction may not equal the scale factor in the negative direction. The scale factors in the two directions must match to avoid rectification. This is also well-known, can be corrected for, but is not relevant to the present invention.
A third source of rectification in a closed-loop capacitive pick-off accelerometer is a dependence of rebalance force on proof mass position. In current art, closed-loop MEMS capacitive pick-off accelerometers with electrostatic feedback use one of two configuration options.
A fourth source of rectification is a force that a damping fluid exerts on the proof mass during vibration. Typically, MEMS accelerometers rely on gas damping to achieve acceptable dynamic performance. Gas-spring damping effects, however, often produce a non-zero time average force on the proof mass as it travels through a cycle of vibration. Many variables affect this rectification error which is a function of the detailed geometry of the damping gaps, the gas type, pressure and temperature, and the magnitude and frequency of the input vibration. The result is a highly complex fluid dynamics problem. Furthermore, the magnitude of this rectification error is potentially extremely large.
Therefore, devices and methods for overcoming these and other limitations of typical state of the art MEMS accelerometers are desirable.
The prior art fails to provide a method for determining a lateral comb drive geometry which minimizes or completely eliminates rectification and, at the same time, provides a sufficiently large force for a given drive area and applied voltage. What is needed in the art is a method for significantly reducing or eliminating this source of rectification within a compact drive geometry.
The method of the present invention provides a geometry of a lateral comb drive for an in-plane, electrostatic force feedback, micromachined accelerometer, closed-loop Coriolis rate gyroscope devices, and closed-loop capacitive pressure and force measuring devices.
When vibration is applied along an accelerometer's input axis, the error in the time-average output, which is the vibration rectification error, due to this input vibration is minimized or eliminated. The geometry resulting from practice of the present invention is space-efficient because drive force is maximized while vibration rectification is minimized or eliminated.
The present invention is an apparatus and method for reducing rectification error in a microelectromechanical system (MEMS) electrostatic comb-drive, closed-loop, in-plane, accelerometer device. This invention provides both an analytical and empirical method of locating a comb tooth overlap that results in minimum or zero rectification error for any chosen general tooth geometry.
The method includes: selecting an initial comb tooth geometry, including selecting initial tooth width and length dimensions, and the spacing between each moveable tooth and the adjacent fixed tooth. These selections are made as a function of overall design requirements and silicon fabrication design rules. A finite element model is constructed of at least one tooth pitch of the initially selected comb tooth geometry. An initial tooth overlap dimension is selected as a starting point. Using a computer aided design (CAD) program, a finite element model is constructed of at least one tooth pitch of the proposed tooth geometry. A finite element analysis calculation of capacitance is performed for both the selected initial tooth overlap dimension and for each of a plurality of different tooth overlap dimensions both greater than and less than the initial tooth overlap dimension. A polynomial fit of at least 4th order is performed for capacitance versus tooth overlap. The 3rd derivative of this polynomial is formed. The tooth overlap dimension that forces this 3rd derivative to zero is then determined. This amount of tooth overlap results in zero vibration rectification.
According to another aspect of the invention, the desired amount of overlap is obtained through an empirical rather than analytical method. A number of test accelerometer devices are fabricated with varying tooth overlap dimensions, and vibration rectification measurements are made. When the range of fabricated geometries span the desired inflection point, interpolation of the test results yields the precise position of zero-rectification.
According to one aspect of the invention, an accelerometer device constructed in accordance with the earlier steps of the method is calibrated to remove rectification errors due to manufacturing variations. These variations include unintended variations that move the ideal inflection point where rectification error is minimum or zero, and geometry variations that introduce a rectification error due to non-symmetries in gas-damping. Two calibration methods are presented. Each method repositions the moveable teeth slightly relative to the fixed teeth, whereby the closed-loop null position is at the actual zero rectification position. Calibrating the accelerometer device is achieved by either bleeding a non-zero bias voltage into the capacitive pick-off circuit of the accelerometer, or by changing one of the two pickoff excitation voltages relative to the other, while vibrating the accelerometer along an input axis thereof. Calibration is complete when the position is found where the resulting measured rectification error is zero. If the nominal comb drive geometry is as determined by the analytical or empirical methods of this invention, then the amount of displacement needed to reposition the moveable teeth to the desired position within an individual accelerometer is very small.
The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same becomes better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:
In the Figures, like numerals indicate like elements.
The Figures illustrate the method of the present invention for determining a lateral comb drive geometry which minimizes or completely eliminates rectification and, at the same time, provides a sufficiently large force for a given drive area and applied voltage.
The example shown in
In Block 201, initial lateral comb tooth geometry: tooth thickness 120 and length 122, the lateral or side-gap 118, and the tooth overlap 114 dimensions, is selected as a function of overall design requirements and conventional fabrication design rules as are generally well-known in the art. Alternatively, the method of the invention is useful for minimizing or eliminating rectification in an existing design. Thus, if actual representative parts are available, then measurements on theses parts provide the tooth geometry. Whether the comb tooth geometry is being currently designed, or is a pre-existing design, the analysis is the same.
In Block 202, a finite element (FE) model is constructed of the initially selected or previously existing lateral comb tooth geometry. For many applications only a small portion of the model, for example one tooth pitch, need be analyzed because symmetry can be applied to determine overall characteristics of the cooperative movable and fixed combs 106, 108. The example in
In Block 203, a finite element analysis (FEA) calculation of capacitance is performed using the CAD program for both the selected initial overlap dimension 114 and for each of many different overlap dimension 114 both greater than and less than the initial overlap dimension 114. As the moveable comb teeth 102 further engages the fixed comb teeth 104, the capacitance between the movable and fixed comb teeth 102, 104 increases monotonically. Therefore, the overlap dimension 114 is modified slightly in the FE model, and the capacitance calculation is performed for this modified overlap dimension 114. This modification of the overlap dimension 114 and performance of the capacitance calculation is repeated for a number of different overlap dimensions around the initial overlap dimension 114. The number of different overlap dimensions 114 utilized, the spacings between adjacent overlap dimensions 114, and the range of different overlap dimensions 114 are determined as a function of the desired accuracy of the result.
Thus, the graph 142 is a plot of capacitance versus the amount of tooth overlap 114 for a small range of engagement. As is well-known in the art, the force between two capacitor surfaces is given by:
where dC/dx is the derivative of capacitance with respect to the overlap distance 114, and V is the electrical potential difference between the two combs 106, 108.
As is observed in the graph 146, the inflection point 126 occurs along the curve of the graphed data, the inflection point 126 occurring at about the point commensurate with the 6.2 micron overlap dimension 114 in this example, where the 2nd derivative of the force relative to position is zero as it transverses between negative and positive values. As discussed above, the 3rd derivative of capacitance relative to position is proportional to the 2nd derivative of the force-displacement relation and so is also zero at the inflection point 126. The smooth behavior and two-fold symmetry of the force-position relationship shown in the graph 146 results in the average force being close to zero over any small region centered about this inflection point 126. That is, the average of a force oscillating about this inflection point 126 is substantially the same as the static value at the inflection point 126. Therefore, the vibration rectification is near zero for overlap dimensions 114 in the vicinity of the inflection point 126.
In Block 204, the capacitance versus position relationship is modeled mathematically using an appropriate mathematical method of curve fitting, such as a polynomial fit. Accordingly, well-known numerical methods are employed for analyzing the capacitance-versus-position relationship and determining a mathematical model of this relationship. When a polynomial fit of the capacitance-versus-position relationship is performed, the polynomial fit is of at least 4th order or at least high enough that residuals from the fit are acceptably small for a given desired accuracy of the result. The 1st derivative of the polynomial is proportional to the force as a function of position.
In Block 205, the 3rd derivative of the capacitance versus position relationship is set to zero and solved for the overlap dimension 114 that forces it to zero. This overlap dimension 114 produces zero rectification. This determination of the zero-rectification overlap dimension 114 completes the definition of the tooth geometry.
Analysis of the exemplary tooth geometry illustrated in
According to this alternative embodiment of the method of the present invention, the tooth length 122 and lateral gap 118 dimensions are optionally allowed to vary. The cooperative fixed and moveable teeth 102, 104 optionally differ from one another in shape. Many geometry variations are optionally allowed, and a “Design of Experiment” is utilized to arrive at a general optimized geometry, of which one goal is low rectification.
In Block 301, initial lateral comb tooth geometry: tooth thickness 120 and length 122, the lateral or side-gap 118, and the tooth overlap 114 dimensions, is selected as a function of overall design requirements and conventional fabrication design rules as are generally well-known in the art.
In Block 302, a series of test specimens of the accelerometer or other device 100 of the invention are fabricated with variations in the tooth overlap 114 within a range of tooth overlap dimensions expected to be experienced in actual operation. The quantity of test specimens fabricated is a function of the desired accuracy of the result.
In Block 303, each test specimen of the accelerometer or other device 100 is placed on a vibration generator or “shaker head,” and the test specimen of the accelerometer or other device 100 is vibrated along it's input axis I (shown in
In Block 304, the rectification data collected in Block 302 is analyzed. When the selected range of tooth overlap dimensions 114 includes the tooth overlap dimension 114 that produces an acceptably low or zero rectification error, the sign of the measured rectification error changes smoothly between negative and positive. This rectification data is interpolated to determine the overlap dimension 114 that produces low or zero rectification error. This determination of the zero-rectification overlap dimension 114 completes the definition of the tooth geometry.
Even with a comb-drive geometry selected per the method of the present invention, natural variation of the manufactured geometry causes a small, non-zero 2nd order electrostatic force-position derivative and, therefore, a rectification error. Manufacturing tolerances also result in slight non-symmetry in damping gap geometry, as discussed below. Therefore, even in the best example of a good design, two sources of small amounts of rectification are present. A net residual rectification is the sum of these two sources of rectification. According to the present invention, the net residual rectification is removed by commanding the proof mass 110 to a null position relative to the frame 112 where the small gas rectification error exactly cancels the error due to a small non-zero 2nd derivative of the electrostatic force-position relationship.
Rectification due to gas-spring effects alone are potentially very large in MEMS accelerometers and other MEMS devices and therefore must be managed to achieve good performance. Therefore, according to one or more different embodiments of the present invention, this gas-spring effects source of rectification is optionally eliminated simultaneously with the rectification source resulting from the manufactured geometry by commanding the proof mass 110 to move to a zero bias-rectification position. Regardless the details of the comb tooth geometry, a gas-damped MEMS device contains small gaps between the moving proof mass 110 and adjacent static parts, i.e., the frame 112. In general, as the proof mass 110 moves in response to external input, some gaps are squeezed smaller while others are increased. The periodic motion of vibration leads to a periodic fluctuation in fluid forces on the proof mass 110. If there is perfect gap symmetry when the proof mass is at its mean position, then the net force on the proof mass 110, averaged over one cycle, is zero. However, due to imperfections in the fabrication of device features, an asymmetry of the gap arrangement will always result. The physics of fluid flow then causes a net time-average force to be applied to the proof mass 110 by the gas. This force cannot be distinguished from an inertial force, and thus represents an error. The present invention provides a method for removing this fluid-driven error due to unintended non-symmetry. Accordingly, in
As discussed above, it is well known in the art of electrostatic accelerometers that the force between the two capacitor plates embodied by the movable and fixed comb drive teeth 102, 104 is directly proportional to the partial derivative of the capacitance with respect to relative motion between the two capacitor plates. The inflection point 126 in
Furthermore, when the net residual rectification error is small enough at the initial null setting of the servo, cancellation alone works well over a narrow range in put amplitudes and frequency.
While the preferred embodiment of the invention has been illustrated and described, it will be appreciated that various changes can be made therein without departing from the spirit and scope of the invention.